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Factor Calculator

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Enter any integer or polynomial expression into the factor calculator above to get a complete list of factors, prime factorization, or factored form. Results appear instantly, with a step-by-step breakdown so you can follow the math.

A factor is a number or expression that divides evenly into another number or expression with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Each of those numbers divides into 12 without leaving anything behind.

Factoring is the reverse of multiplication. Instead of combining numbers to get a product, you break a product apart into the pieces that were multiplied together.

This free factoring calculator handles:

  • Whole numbers and large numbers
  • Prime factorization
  • Polynomials, including quadratic expressions
  • Greatest common factor (GCF) and least common multiple (LCM)

How to Use the Factoring Calculator Step by Step

  1. Choose what you want to calculate. Select whether you are factoring a number or a polynomial expression.
  2. Enter your value. Type an integer (like 91) or a polynomial (like x² + 5x + 6) into the input field.
  3. Click "Calculate." The calculator returns all factors, prime factors, or the factored polynomial form.
  4. Read the step-by-step solution. Each result includes the method used so you can verify or learn the process.

That's it. No sign-up required.

Find the Factors of a Number

The factors of a number are all the integers that divide evenly into it. To find them, test each integer starting from 1 up to the number itself. Every time division produces no remainder, you have found a factor pair.

What are the factors of 24? 1, 2, 3, 4, 6, 8, 12, and 24. Each divides into 24 with zero remainder. The factor pairs are (1, 24), (2, 12), (3, 8), and (4, 6).

What are the factors of 91? 1, 7, 13, and 91. Since 91 = 7 × 13, it has only four factors.

Can a number have an odd number of factors? Yes. Perfect squares always have an odd number of factors because one factor pair contains the same number twice. For example, 36 has factors 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine factors total) because 6 × 6 = 36.

What about negative numbers? Every positive factor has a negative counterpart. The factors of 12 technically include both 1 and negative 1, both 2 and negative 2, and so on. Most factor calculators list only the positive factors unless you specify otherwise.

Factor Numbers and Large Numbers

For small numbers, you can test divisors by hand. For large numbers, this process can become quite tedious. The calculator uses trial division and other efficient methods to factor numbers quickly, even when they reach into the millions.

When do you need to factor numbers?

  • Simplifying fractions
  • Finding common denominators
  • Solving algebraic equations
  • Problems in number theory and cryptography (where large prime factors power modern encryption)

Prime Factorization

A prime number has only two factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11, 13.

Prime factorization breaks a number down into the prime factors that, when multiplied together, produce the original number. Every integer greater than 1 has a unique prime factorization.

Example: The prime factorization of 60 is 2 × 2 × 3 × 5, or 2² × 3 × 5.

What are prime factors? They are the prime numbers that divide evenly into a given number. For 60, the prime factors are 2, 3, and 5.

What are the factors of a prime number? Just 1 and itself. That's the definition. If a number has any other factor, it is not prime.

Factoring Polynomials

Factoring polynomials means rewriting an algebraic expression as a product of simpler expressions. This is a core skill in algebra because it lets you solve equations, simplify fractions, and analyze graphs.

A polynomial like x² + 5x + 6 factors into (x + 2)(x + 3). You can verify by multiplying the two binomials back together.

How do you factor a monomial? Pull out the greatest common factor from the coefficient and variables. For example, 6x³ = 2 × 3 × x × x × x.

How do you factor a binomial? Look for a common factor first. Then check for special patterns like the difference of squares (a² − b² = (a + b)(a − b)) or the sum/difference of cubes.

How do you factor a trinomial? For a quadratic trinomial ax² + bx + c, find two numbers that multiply to a × c and add to b. Split the middle term and factor by grouping. The calculator automates this process and shows each step.

Use a Factor Calculator to Find Polynomial Factors

Factoring by hand works well for simple expressions. When the coefficients get larger or when you are unsure which method applies, use this factor calculator to find the factored form instantly.

The tool handles:

  • Quadratic expressions (ax² + bx + c)
  • Factor by grouping
  • Difference of squares, sum of cubes, and difference of cubes
  • Expressions with common monomial factors

What if the quadratic does not factor nicely with integers? Some quadratics have irrational or complex roots. In that case, the quadratic formula gives exact solutions, but the expression cannot be factored over the integers. The calculator will indicate when a clean integer factorization is not possible.

When can a quadratic be factored over the real numbers? When the discriminant (b² − 4ac) is greater than or equal to zero. If the discriminant is a perfect square, the quadratic factors with rational numbers.

Key Formulas for Factorization

Formula for Prime Factorization

To find the prime factorization of a given integer:

  1. Divide the number by the smallest prime (2).
  2. If it divides evenly, record that prime factor and divide again.
  3. When it no longer divides evenly, move to the next prime (3, then 5, 7, 11, and so on).
  4. Continue until the quotient is 1.

Example: Prime factorization of 180

  • 180 ÷ 2 = 90
  • 90 ÷ 2 = 45
  • 45 ÷ 3 = 15
  • 15 ÷ 3 = 5
  • 5 ÷ 5 = 1

Result: 180 = 2² × 3² × 5

This trial division method works reliably for any number. For very large numbers, more advanced algorithms exist, but the principle is the same.

Formula for Factoring Polynomials

Difference of squares: a² − b² = (a + b)(a − b)

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)

Quadratic trinomial (a = 1): x² + bx + c = (x + m)(x + n), where m × n = c and m + n = b

General quadratic (a ≠ 1): Use the ac method. Find two numbers that multiply to a × c and add to b, then factor by grouping.

These key formulas cover the vast majority of factoring problems you will encounter in algebra courses and standardized tests.

Step-by-Step: How to Calculate and Factor

Using the Factoring Calculator to Factor Numbers

  1. Simply enter an integer into the calculator (for example, 84).
  2. The tool lists the factors of the number: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
  3. It also displays the prime factorization: 2² × 3 × 7.
  4. Factor pairs appear as well: (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12).

Use this output to simplify fractions, find the GCF of two numbers, or check homework.

What is the difference between factors and multiples? Factors divide into a number. Multiples are produced by multiplying a number. The factors of 6 are 1, 2, 3, 6. The multiples of 6 are 6, 12, 18, 24, and so on.

Using the Factoring Calculator to Factor Polynomials

  1. Enter a polynomial expression, such as 2x² + 7x + 3.
  2. The calculator identifies the type of expression and selects the appropriate method.
  3. It returns the factored form: (2x + 1)(x + 3).
  4. A step-by-step breakdown shows how the middle term was split and how grouping was applied.

How do you solve factoring by greatest common monomial factor? Before applying any special formula, always check whether all terms share a common factor. For 6x³ + 9x², the GCF is 3x², giving 3x²(2x + 3).

Factor Calculator to Find Greatest Common Divisor and Least Common Multiple

How do you find the greatest common factor (GCF)? The GCF (also called the greatest common divisor, or GCD) of two numbers is the largest number that divides evenly into both. Use prime factorization for each number, then multiply the common prime factors together.

Example: GCF of 36 and 48.

  • 36 = 2² × 3²
  • 48 = 2⁴ × 3
  • Common primes at their lowest powers: 2² × 3 = 12

The GCF is 12.

How do you find the least common multiple (LCM) using factors? The LCM is the smallest number that is a multiple of both numbers. Use the prime factorizations and take each prime factor at its highest power.

Example: LCM of 36 and 48.

  • Take 2⁴ × 3² = 144

The LCM is 144.

How to find LCM with the listing multiples method? List multiples of each number until you find the first one they share. Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... The LCM is 12. This works for small numbers but becomes tedious for larger ones, which is why the prime factorization method (or this calculator) is more practical.

You can also use the relationship: LCM(a, b) = (a × b) ÷ GCF(a, b).

Factorization of Large Numbers

Factoring large numbers is computationally harder than factoring small ones. Trial division still works, but it gets slow as numbers grow past six or seven digits.

This matters beyond math class. Modern encryption (like RSA) relies on the difficulty of factoring very large numbers into their prime factors. The security of online transactions depends on this mathematical challenge.

The calculator handles large numbers efficiently using optimized trial division. For most practical purposes (homework, number theory exercises, problem-solving), it returns results in under a second.

Tips for factoring large numbers by hand:

  • Check divisibility by 2 (even number), 3 (digit sum divisible by 3), and 5 (ends in 0 or 5) first.
  • You only need to test primes up to the square root of the number.
  • If no prime up to the square root divides evenly, the number itself is prime.

Factors of a Number in Algebra and Equations

Factoring connects arithmetic to algebra. When you factor a quadratic equation like x² − 5x + 6 = 0 into (x − 2)(x − 3) = 0, you can immediately find the solutions: x = 2 and x = 3.

Why is finding factors important in maths?

  • It simplifies algebraic expressions and makes equations solvable.
  • It reduces fractions to lowest terms.
  • It reveals structure in number theory and mathematical proofs.
  • It supports problem-solving across geometry, calculus, and applied math.

Numbers have factors. Algebraic expressions have factors. The concept is the same: break a mathematical object into the pieces that, when multiplied together, rebuild the original.

What is the difference between factors and multiples in algebra? Factors of an expression divide into it evenly. Multiples of an expression are produced by multiplying it by another expression. The factor (x + 2) divides into x² + 5x + 6. The expression 3(x + 2) = 3x + 6 is a multiple of (x + 2).

This calculator is a free planning and learning aid. For graded coursework or professional applications, verify results independently or consult a qualified instructor.